from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,18,5]))
pari: [g,chi] = znchar(Mod(853,1001))
Basic properties
| Modulus: | \(1001\) | |
| Conductor: | \(1001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1001.cw
\(\chi_{1001}(83,\cdot)\) \(\chi_{1001}(216,\cdot)\) \(\chi_{1001}(398,\cdot)\) \(\chi_{1001}(447,\cdot)\) \(\chi_{1001}(580,\cdot)\) \(\chi_{1001}(629,\cdot)\) \(\chi_{1001}(811,\cdot)\) \(\chi_{1001}(853,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((430,365,925)\) → \((-1,e\left(\frac{9}{10}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 1001 }(853, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)